Question

The summation of two numbers is 10 and cube difference is 2. Then what will be the big number between them?

Answer 1

Solution:

Let the two numbers are x, y

Then x + y = 10

x³ - y³ = 2

or, x³ - (10 - x)³ = 2

or, x³ - (1000 - 300x + 30x² - x³) - 2 = 0

or, x³ - 1000 + 300x - 30x² + x³ - 2 = 0

or, 2x³ - 30x² + 300x - 1002 = 0

or, x³ - 15x² + 150x - 501 = 0 ... (1)

To solve equation no. (1), we first remove the seconds term in order to reduce it to the standard form ( z³ + 3Hz + G = 0 ) for Cardan's method solution.

Let x = w + h

Then (1) no. equation becomes

(w + h)³ - 15 (w + h)² + 150 (w + h) - 501 = 0

or, w³ + (3h - 15) w² + (3h² - 30h + 150) w + (h³ - 15h² + 150h - 501) = 0

So h = 5 [ taking 3h - 15 = 0 ] reduces the equation to

w³ + 75w - 1 = 0 ... (2)

Now this equation can be solved using Cardan's method.

Let w = u + v

Then w³ = (u + v)³

or, w³ = (u³ + v³) + 3uv (u + v)

or, w³ = (u³ + v³) + 3uvw

or, w³ - 3uvw - (u³ + v³) = 0

Comparing it with equation no. (2), we get

uv = - 25 and u³ + v³ = 1

Now we find the values of u and v in real numbers.

Comparing (2) no. equation with the standard form

z³ + 3Hz + G = 0, we get

H = 25 and G = - 1

Then u³ = 1/2 * [H + √(G² + 4H³)]

or, u = 4.99333 [ we take the real value of u only ]

Then v = - 25/u

or, v = - 5.00668

Therefore, x = w + h = u + v + h

or, x = 4.99333 - 5.00668 + 5

or, x = 4.98665

So y = 10 - 4.98665

or, y = 5.01335

Therefore, the two numbers are 4.98665 and 5.01335, of which 5.01335 > 4.98665 .

Note:

1. Difference is considered to be in magnitude so, whatever value we calculate for x and y, x > y or x < y in considerations don't matter because this is a problem containing equations.

2. Cardan's method is in higher level syllabus so there might be a problem to solve the equation for school level students. However the problem is solved in easy calculations.

3. For this problem, we take the real values of u, v in order to exclude other imaginary solutions for x, y.

4. Values of u, v, x, y and others are in approximation values to ignore the decimal values for a large number of digits.